# Dervie combined couette - poiseuille flow using non dimesionalized fundamental equations of compressible viscous flow

When I try to derive the combined Couette - poiseuille flow using non-dimensionalized fundamental equations of compressible viscous flow, I get pressure gradient as a function of both x and y

$$Non-dimensionalized$$ $$Navier-Strokes$$ $$Equation$$

in x-direction

$$\frac{\partial u^*}{\partial t^*} + u^*\frac{\partial u^*}{\partial x^*}+v^*\frac{\partial u^*}{\partial y^*} = -\frac{\partial p^*}{\partial x^*}+\frac{1}{Re}(\frac{\partial^{2} u^*}{\partial {x^*}^{2}}+\frac{\partial^{2} u^*}{\partial {y^*}^{2}})+\frac{1}{{Fr}^{2}}\$$

in y-direction

$$\frac{\partial v^*}{\partial t^*} + u^*\frac{\partial v^*}{\partial x^*}+v^*\frac{\partial v^*}{\partial y^*} = -\frac{\partial p^*}{\partial y^*}+\frac{1}{Re}(\frac{\partial^{2} v^*}{\partial {x^*}^{2}}+\frac{\partial^{2} v^*}{\partial {y^*}^{2}})+\frac{1}{{Fr}^{2}}\$$

Where $$Re$$ is the Reynolds number, and $$Fr$$ is the Froude number.

Now from our assumptions that the flow is

2. fully developed( variation of velocity in the x direction is zero)
3. incompressible
4. Newtonian fluid

the above equations can be re-written as

in x- direction

$$-\frac{\partial p^*}{\partial x^*}+ \frac{1}{Re}(\frac{\partial^{2} u^*}{\partial {y^*}^{2}}) + \frac{1}{{Fr}^{2}}\ = 0$$

in y- direction

$$-\frac{\partial p^*}{\partial y^*}+ \frac{1}{{Fr}^{2}}\ = 0$$

Which results in the pressure gradient being a function of both x and y, which is not true since, from the poiseuille flow, we know that the pressure gradient is only a function of x

Please explain how can I derive the combined flow using non dimensionalized equations

My reputation isn't high enough to write this as a comment. However, often one assumes in flows such as Couette, Poiseuille, or Couette-Poiseuille that the fluid isn't being acting under gravity. In such a case, the reciprocal of the Froude number is zero and hence $$-\frac{\partial p^*}{\partial y}=0$$, so that the pressure gradient is only a function of $$x$$.